Flop bet should be around 25% pot.
Turn you can't check behind since you are OOP,. You are not getting called by worse especially with Qh blocking the main FD you beat. Trivial check and probably c/f against under bluffing microstakes pool.

I agree, turn is a check and I dont really understand the shove. What is the reason behind the shove? I think this is the most important question. Is it for value or bluff?

It's hard to pin down opponents ranges in this spot. I think we mostly have QQ+, AK, some AQs and KQs and maybe A5s as our cold 4b bluff??? Not sure that will be in there. We are more concerned about his range here. I think QQ-TT are almost always in there, AA/KK at some small frequency, AK, maybe some AQ, maybe some KQ, Some Axs.

I usually simplify this spot and just jam flop with QQ/KK here and check AA and JJ (if I have it). I would also just jam my NFD's (if they are even in my range).

If we employ a 3 option strategy, can't really go wrong either betting smallish or ripping it in.

Keeping ranges the same, there is very little EV loss (as shown below) employing a Jam strategy here. These ranges also assume that our opponent is playing perfectly and has the ranges I used. If they have more Ax/Kx in their range, equity denial becomes even more incentivized.

Overall EV and EV of QQ Jam

Overall EV and EV of QQ Two Bet Sizes

Checking Turn is preferable over betting big/jamming

You can actually bet small and fold to raise on turn.